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Difficulty: Very hardFactors, Multiples, and Primes

The greatest common factor (GCF) of two positive integers, xx and yy, is 1515, and their least common multiple (LCM) is 3,1503,150. If x=150x = 150, which of the following represents the prime factorization of yy?

  1. 32×5×73^2 \times 5 \times 7Answer
  2. B
    2×3×5×72 \times 3 \times 5 \times 7
  3. C
    3×73 \times 7
  4. D
    2×32×52×72 \times 3^2 \times 5^2 \times 7

Answer

The prime factorization of y is 32×5×73^2 \times 5 \times 7.
The correct answer can be found by comparing prime exponents. The exponent of a prime in the GCF is the minimum of its exponents in xx and yy, while in the LCM it is the maximum. Given x=21×31×52x = 2^1 \times 3^1 \times 5^2, GCF=20×31×51\text{GCF} = 2^0 \times 3^1 \times 5^1, and LCM=21×32×52×71\text{LCM} = 2^1 \times 3^2 \times 5^2 \times 7^1. For prime 2, the min is 0, so yy must have 202^0. For prime 3, the max is 2, so yy must have 323^2. For prime 5, the min is 1, so yy must have 515^1. For prime 7, the max is 1, so yy must have 717^1. Combining these yields 32×5×73^2 \times 5 \times 7.

Step-by-Step Solution

1
Find the prime factorizations of xx, the GCF, and the LCM.
x=150=21×31×52x = 150 = 2^1 \times 3^1 \times 5^2
GCF=15=20×31×51\text{GCF} = 15 = 2^0 \times 3^1 \times 5^1
LCM=3,150=21×32×52×71\text{LCM} = 3,150 = 2^1 \times 3^2 \times 5^2 \times 7^1
Comparing the exponents of prime factors between the numbers, their GCF, and their LCM allows us to deduce the exact prime factorization of yy.
2
Determine the necessary exponent for the prime factor 22 in yy.
The exponent for 22 in yy must be 00.
The GCF exponent is the minimum between xx and yy. Since the GCF has 202^0 and xx has 212^1, yy must provide the minimum of 00, meaning yy does not contain a factor of 22.
3
Determine the necessary exponent for the prime factor 33 in yy.
The exponent for 33 in yy must be 22.
The LCM exponent is the maximum between xx and yy. Since the LCM has 323^2 and xx has 313^1, yy must provide the maximum of 22, so it has 323^2.
4
Determine the necessary exponents for the prime factors 55 and 77 in yy.
The exponent for 55 is 11, and the exponent for 77 is 11.
For 55: The GCF has 515^1 and xx has 525^2, so yy must provide the minimum 11. For 77: The LCM has 717^1 and xx has 707^0, so yy must provide the maximum 11.
5
Combine the deduced prime factors to form the final prime factorization of yy.
y=20×32×51×71=32×5×7y = 2^0 \times 3^2 \times 5^1 \times 7^1 = 3^2 \times 5 \times 7
Multiplying the prime powers gives the complete prime factorization of yy.

Key Concept

Using prime factorizations to determine relationships between numbers, their Greatest Common Factor (GCF), and their Least Common Multiple (LCM).

Alternative Method

Instead of using the prime factorization minimum/maximum rules, you can use the mathematical property that the product of two positive integers is equal to the product of their GCF and LCM. Therefore, x×y=GCF×LCMx \times y = \text{GCF} \times \text{LCM}. Substituting the given values: 150×y=15×3,150150 \times y = 15 \times 3,150. This simplifies to 150y=47,250150y = 47,250. Solving for yy gives y=315y = 315. Finding the prime factorization of 315315 yields 32×5×73^2 \times 5 \times 7.
Estimated Time:2m 0s
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Topics

Factors, Multiples, and Primes

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